Using recent results on dynamics of non-stationary random matrix products, we establish spectral and dynamical localization for 1D Schrodinger operators with potentials given by independent but not identically distributed random variables.
We consider the problem of classifying Kolmogorov automorphisms (or K-automorphisms for brevity) up to isomorphism or up to Kakutani equivalence. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and K-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. Therefore one might hope to extend Ornstein’s classification of Bernoulli shifts up to isomorphism by a numerical Borel invariant to a classification of K-automorphisms by some type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to K-automorphisms, considered as a subset of the Cartesian product of the set of K-automorphisms with itself, is a complete analytic set, and hence not Borel. Moreover, we prove this remains true if we restrict consideration to K-automorphisms that are also C∞ diffeomorphisms. In addition, all of our results still hold if “isomorphism” is replaced by “Kakutani equivalence”. This shows in a concrete way that the problem of classifying K-automorphisms up to isomorphism or up to Kakutani equivalence is intractable. These results are joint work with Philipp Kunde.
I will give an overview of the most important results about stationary random walks on SL(k, R). We will talk about Lyapunov exponent and their properties, such as positivity of the top exponent, simplicity and regularity of the spectrum and others. We will also mention other limit theorems, such as central limit theorem and law of iterated logarithm. The talk is based on the monograph ``Random walks on groups and random transformations'' by Alex Furman.
A variety of questions and results on Cantor sets revolved around the Minkowski sums of Cantor sets and the topological structure or Hausdorff dimension of these sumsets. For example, Shmeling and Schmerkin showed that given an increasing sequence {x_n} bounded by 0 and 1, there exists a Cantor set C such that x_n is the Hausdorff dimension of C added to itself n times.
Given any integer n, we will provide a construction for a Cantor set with zero logarithmic capacity such that the Cantor set added to itself n times is a single interval, while a sum of any smaller number of copies of that set is still a Cantor set.
We will give a survey of various results concerning the stable leaves of hyperbolic systems, particularly over smooth surfaces. We will discuss the regularity of foliations that the stable leaves form or can be included into.
For this last talk in the series, I will discuss the details of how spectrum of the Schrödinger operator can be defined, and the proof that the spectrum can consist of an infinite number of intervals.
A set of matrices can be defined as uniformly hyperbolic if products of the matrices have a norm that grows exponentially. A paper written by Avila, Bocci, and Yoccoz in 2008 has expanded on this concept and posed a variety of questions on this subject. In this talk we will go over some of the concepts covered in this paper, a few additional tools developed to help study this subject, and ways the tools are being used to address the questions posed.
We begin by discussing a conjectured version of a non-stationary stable manifold theorem for a hyperbolic horseshoe and small perturbations of it, and discuss the techniques needed to achieve such a result. With these techniques in mind, we will conjecture a similar result for the trace maps on a particular family of cubic surfaces, and explain what deductions this result would make about the spectrum of related discrete Schrodinger operators with Sturmian potential.
We will prove that there exists a discrete Schrodinger operator with a potential given by a sum of a random potential and a periodic background, with the spectrum that consists of an infinite number of intervals.
A hyperbolic locus $\mathcal{H} \subset SL(2,R)^n$ is a connected open set such that for all $x\in\mathcal{H}$, $\{x_i\}_1^n$ is a uniformly hyperbolic set of matrices. In $SL(2,R)^2$, the geometry of the loci was studied in Avila, Bochi, and Yoccoz's 2008 work. In this talk, some of the details of the geometry in higher dimensions will be discussed, as well as the relevance with Schrodinger operators.